Download Classical Chaos in Geometric Collective Model

CHAOTIC DYNAMICS IN COLLECTIVE
MODELS OF NUCLEI
Pavel Stránský1,2
1Institute
of Particle and Nuclear Phycics
Faculty of Mathematics and Physics
Charles University in Prague, Czech Republic
2Instituto
de Ciencias Nucleares
Universidad Nacional Autonoma de México
Collaborators: Michal Macek1, Pavel Cejnar1
Alejandro Frank2, Ruben Fossion2, Emmanuel Landa2
XXXIII Symposium on Nuclear Physics, Cocoyoc, Mexico, 2010
5th January 2010
CHAOTIC DYNAMICS IN COLLECTIVE
MODELS OF NUCLEI
Introduction
- Basics of Geometric Collective Model (GCM)
(restricted to nonrotating motions)
1. Classical chaos in GCM
- Measures of regularity
- Geometrical method
2. Quantum chaos in GCM
- Short-range correlations and Brody parameter
- Peres lattices
- Long-range correlations and 1/f noise
- Comparison of classical and quantum dynamics
3. Interacting Boson Model (IBM)
- Application of the above mentioned methods
Geometrical Collective Model
(restricted to nonrotating motions)
Introduction: Geometric Collective Model
Hamiltonian of GCM
T…Kinetic term
V…Potential
Neglect higher order terms
Corresponding tensor of momenta
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
Principal Axes System
Shape variables:
B … strength of nonintegrability
(B = 0 – integrable quartic oscillator)
neglect
Introduction: Geometric Collective Model
Hamiltonian of GCM
T…Kinetic term
V…Potential
Neglect higher order terms
Corresponding tensor of momenta
neglect
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
Principal Axes System
Phase structure
B
V
V
A
b
C=1
Deformed shape
Spherical shape
b
Introduction: Geometric Collective Model
Hamiltonian of GCM
Principal Axes System
Classical dynamics
Nonrotating case J = 0!
– Hamilton equations of motion
Quantization
– Diagonalization in oscillator basis
2 physically important quantization options
(with the same classical limit):
• oportunity to test Bohigas conjecture for different quantization schemes
(a) 5D system restricted to 2D
(true geometric model of nuclei)
(b) 2D system
1. Classical chaos in GCM
1. Classical chaos in GCM
Fraction of regularity
Measure of classical chaos
Poincaré section
vx
x
REGULAR area
CHAOTIC area
freg=0.611
A = -1, C = K = 1
B = 0.445
1. Classical chaos in GCM
Different definitons & comparison
Surface of chosen
Poincaré section
number of

regular
total
trajectories
(with random
initial conditions)
Statistical
measure
E=0
control
parameter
1. Classical chaos in GCM
Complete map of classical chaos in GCM
chaotic
Shape-phase transition
regularity”
Integrability
Veins of
regularity
regular
“Arc of
control
parameter
Global minimum
and saddle point
Region of phase
transition
1. Classical chaos in GCM
Geometrical method
Hamiltonian in flat Eucleidian space with potential:
Conformal
metric
Hamiltonian of free particle in curved space:
Application of methods of Riemannian geometry
Negative eigenvalues of the matrix
inside kinematically accesible area induce nonstability.
L. Horwitz et al., Phys. Rev. Lett. 98 (2007), 234301
1. Classical chaos in GCM
Geometrical method
(a)
(b)
(c)
(d)
(d)
(c)
y
x
(b)
(a)
Global minimum
and saddle point
Convex-Concave transition
Region of phase
transition
2. Quantum chaos in GCM
2. Quantum chaos in GCM
Spectral statistics
Nearestneighbor
spacing
distribution
P(s)
Poisson
GOE
s
REGULAR system CHAOTIC system
Brody
distribution
parameter w
- Artificial interpolation between Poisson and GOE distribution
- Measure of chaoticity of quantum systems
- Tool to test classical-quantum correspondence
Bohigas conjecture (O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1)
2. Quantum chaos in GCM
Peres lattices
Quantum system:
Infinite number of of integrals of
motion can be constructed
(time-averaged operators P):
Lattice: energy Ei versus value of
Integrable
lattice always ordered
for any operator P
nonintegrable
B=0
partly ordered,
partly disordered
B = 0.445
<P>
<P>
regular
E
E
regular
chaotic
A. Peres, Phys. Rev. Lett. 53 (1984), 1711
2. Quantum chaos in GCM
Hamiltonian of GCM
Principal Axes System
Nonrotating case J = 0!
H’
Independent
Peres operators in GCM
L22D
L25D
(a) 5D system restricted to 2D
(true geometric model of nuclei)
(b) 2D system
2. Quantum chaos in GCM
Nonintegrable perturbation
Small perturbation affects only localized part of the lattice
B=0
B = 0.005
B = 0.05
B = 0.24
<L2>
Remnants of
regularity
<H’>
E
Integrable
Increasing perturbation
Empire of chaos
2. Quantum chaos in GCM
“Arc of regularity” B = 0.62
<L2>
<VB>
2D
(different quantizations)
5D
E
2. Quantum chaos in GCM
“Arc of regularity” B = 0.62
<L2>
<VB>
2D
(different quantizations)
5D
E
• Connection with the arc of
regularity (IBM)
• b – g vibrations resonance
2. Quantum chaos in GCM
Dependence on the classicality parameter
<L2>
Zoom into sea of levels
E
2. Quantum chaos in GCM
Classical and quantum
measure - comparison
B = 0.24 Classical measure
B = 1.09
Quantum measure (Brody)
2. Quantum chaos in GCM
1/f noise
- Fourier transformation of the time series
Power spectrum
a=1
CHAOTIC system
a=2
REGULAR system
- Direct comparison of
- In GCM we cannot average over ensembles!!!
A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)
2. Quantum chaos in GCM
1/f noise
Integrable case: a = 2 expected
(4096 levels starting from level 2000)
A = -1
A = +1
2. Quantum chaos in GCM
1/f noise
Comparison of measures
B = 0.24
B = 0.62
3. Chaos in IBM
3. Interacting Boson Model
3. Chaos in IBM
IBM Hamiltonian
a – scaling parameter
3 different dynamical
symmetries
O(6)
0
0
Invariant of O(5)
(seniority)
1
Casten triangle
SU(3)
U(5)
3. Chaos in IBM
IBM Hamiltonian
a – scaling parameter
3 different dynamical
symmetries
3 different
Peres operators
O(6)
0
0
Invariant of O(5)
(seniority)
1
Casten triangle
SU(3)
U(5)
3. Chaos in IBM
Regular lattices in integrable case
- even the operators non-commuting with
Casimirs of U(5) create regular lattices !
commuting
40
30
non-commuting
0
n̂d
20
-10
U(5)
10
SU
SU33
ˆ
ˆ
Q
.
Q
ˆ
ˆ
Q.Q
-20
-30
limit
0
-40
0
vv
n̂d
-10
N = 40
L=0
-20
-30
-40
Qˆ .Qˆ O 6
3. Chaos in IBM
Different invariants
classical
regularity
Arc of regularity
h = 0.5
N = 40
U(5)
SU(3)
O(5)
3. Chaos in IBM
Different invariants
Arc of regularity
<L2>
GOE
classical
regularity
h = 0.5
N = 40
U(5)
SU(3)
O(5)
3. Chaos in IBM
Application: Rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
n̂d
n̂d
N = 30
L=0
Qˆ .Qˆ SU 3
3. Chaos in IBM
Application: Rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
n̂d
N = 30
L = 0,2
Qˆ .Qˆ SU 3
3. Chaos in IBM
Application: Rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
n̂d
N = 30
L = 0,2,4
Qˆ .Qˆ SU 3
3. Chaos in IBM
Application: Rotational bands
η = 0.5, χ= -1.04 (arc of regularity)
n̂d
N = 30
L = 0,2,4,6
Qˆ .Qˆ SU 3
This is the last slide
Summary
Thank you for
your attention
1. Collective models of nuclei
•
•
Complex behavior encoded in simple dynamical
equation
Possibility of studying manifestations of both
classical and quantum chaos and their relation
2. Peres lattices
•
•
•
Allow visualising quantum chaos
Capable of distinguishing between chaotic and
egular parts of the spectra
Freedom in choosing Peres operator
3. Methods of Riemannian geometry
•
Determine location of the onset of chaoticity in
classical systems
4. 1/f Noise
•
Preliminary results, deeper investigation should
be done
http://www-ucjf.troja.mff.cuni.cz/~geometric
~stransky
1. Classical chaos in GCM
How to distinguish quasiperiodic and
unstable trajectories numerically?
1. Lyapunov exponent
Divergence of two neighboring trajectories
2. SALI (Smaller Alignment Index)
• two divergencies
• fast convergence towards zero for chaotic trajectories
Ch. Skokos, J. Phys. A: Math. Gen 34 (2001), 10029; 37 (2004), 6269
2. Quantum chaos in GCM
Wave functions
<L2>
<VB>
E
Probability
densities
regular
chaotic
regular
2. Quantum chaos in GCM
Wave functions and Peres lattice B = 1.09
OT
E
Peres lattice
E
convex → concave
(regular → chaotic)
b
Probability
density of wave
functions
2. Quantum chaos in GCM
Long-range correlations
• Short-range correlations – nearest neighbor spacing distribution
• number variace
• D3 („spectral rigidity“)
Only 1 realization of the ensemble in GCM – averaging impossible
Chaoticity of the system changes with energy – nontrivial
dependence on both L and E